Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-x+8y &= 9 \\ 2x+4y &= 7\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}x-8y &= -9\\ 4x+8y &= 14\end{align*}$ Add the top and bottom equations. $5x = 5$ Divide both sides by $5$ and reduce as necessary. $x = 1$ Substitute $1$ for $x$ in the top equation. $- 1+8y = 9$ $-1+8y = 9$ $8y = 10$ $y = \dfrac{5}{4}$ The solution is $\enspace x = 1, \enspace y = \dfrac{5}{4}$.